Can be ζ(3) written as απβ, where (α,β∈C), β≠0 and α doesn’t depend of π (like √2, for example)?
Several ζ values are connected with π, like:
and so on for all even numbers.
See this mathworld link to more details: Riemann Zeta Function
So the question is, could ζ(3) be written as:
α not dependent of π
See α not essencially belongs Q and α,β could be real numbers too.
When I wrote α is not dependent of π it’s a strange and a hard thing to be defined, but maybe α can be written using e or γ or √2 or some other constant.
Maybe this still a open question. If
∑2k=0(−1)kB2k B2−2k+2(2k)! (2−2k+2)!
in −4∑2k=0(−1)kB2k B2−2k+2(2k)! (2−2k+2)!π3 be of the form δπ3 with δ not dependent of π
and −2∑k≥1k−3e2πk−1 not dependent of π too, this question still hard and open.
I discovered a result, but later I’ve seen that this is something already known, either way, it is an interesting one to have it here.
but, if ζ′(−2) is of the form απ2, with α not dependent of π, then this still remains as a hard and an open question.
I have a conjecture that ζ′(−2) will not cancel the π2 term, but since I wasn’t able to prove it and I can’t use it here.
We can express ζ of odd numbers with ζ′ in a easy way, with a “closed” form like this one.
The question is whether or not ζ(3) is connected with π. The answer is yes. Moreover, ζ(3)=βπα for some complex α,β. Take α=3 and β=0.0387682…. It is not known whether β=0.0387682… is algebraic or transcendental. It is known, however, that ζ(3) is irrational as shown by Apery.
Ramanujan conjectured and Grosswald proved that the following holds. If α,β>0 such that αβ=π2, then for each non-negative integer n,
22nn+1∑k=0(−1)kB2k B2n−2k+2(2k)! (2n−2k+2)!αn−k+1βk.
where Bn is the nth-Bernoulli number.
For odd positive integer n, we take α=β=π,
ζ(2n+1)=−22n(n+1∑k=0(−1)kB2k B2n−2k+2(2k)! (2n−2k+2)!)π2n+1−2∑k≥1k−2n−1e2πk−1.
In particular, for n=1,
ζ(3)=−4(2∑k=0(−1)kB2k B2−2k+2(2k)! (2−2k+2)!)π3−2∑k≥1k−3e2πk−1.
Observe that the coefficient of π3 is rational, however, nothing is known about the algebraic nature of the infinite sum. This is a current topic of research. Indeed, it is conjectured that ζ(3)π3 is transcendental.
Update: Recently, Takaaki Musha claims to have proved that ζ(2n+1)(2π)2n+1 is irrational for positive n≥1. However, some objection has since been raised (read comments below).